Kol's evolution equations tested episode by episode and the structure of the absorptivity

We test the ingredients of Kol's statistical evolution equations for the planar three-body problem with zero total angular momentum, dividing 24,000 free falls into 110,117 chaotic episodes separated by excursions. The excursions that return are reabsorbed by chaos with the same absorptivity measured by external launch, from 0.06 to 0.23 depending on the channel. The exits from chaos have almost no memory, and the escape probabilities with unequal masses are predicted within the statistical error, with a mean error of 0.6 to 0.9 percentage point, when the absorptivity and the measurement use the same definition of chaotic episode; the residual of the previous note came from inconsistent definitions. The dependence of the outcomes on the lifetime comes from the first episode and from the regular escapes, two channels predicted by the theory. The duration of the chaotic episodes, however, is not exponential. The absorptivity does not follow a simple law of the kick and the encounter speed, which explain at most 71% of its variation and fail for masses not used in the fit. Maps of the first passage show smooth regular bands, excursion zones that are also smooth and concentrate most of the absorptivity, and a small direct chaotic set with a fractal boundary. The absorptivity of a whole run equals the probability that the first passage ends in excursion or in chaos (correlation of 0.98 in 147 situations), which reduces the problem to a single encounter; in it, the choice between flyby and exchange depends almost only on the masses, but there is not yet a universal law.

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-04
DOI
https://doi.org/10.5281/zenodo.23964547
Primary Topic
Quantum chaos and dynamical systems
Type
article
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article

Kol's evolution equations tested episode by episode and the structure of the absorptivity

Willian Wistuba Melo da Cunha
Zenodo (CERN European Organization for Nuclear Research)
Quantum chaos and dynamical systems
article

Kol's evolution equations tested episode by episode and the structure of the absorptivity

Willian Wistuba Melo da Cunha
article en

Abstract

We test the ingredients of Kol's statistical evolution equations for the planar three-body problem with zero total angular momentum, dividing 24,000 free falls into 110,117 chaotic episodes separated by excursions. The excursions that return are reabsorbed by chaos with the same absorptivity measured by external launch, from 0.06 to 0.23 depending on the channel. The exits from chaos have almost no memory, and the escape probabilities with unequal masses are predicted within the statistical error, with a mean error of 0.6 to 0.9 percentage point, when the absorptivity and the measurement use the same definition of chaotic episode; the residual of the previous note came from inconsistent definitions. The dependence of the outcomes on the lifetime comes from the first episode and from the regular escapes, two channels predicted by the theory. The duration of the chaotic episodes, however, is not exponential. The absorptivity does not follow a simple law of the kick and the encounter speed, which explain at most 71% of its variation and fail for masses not used in the fit. Maps of the first passage show smooth regular bands, excursion zones that are also smooth and concentrate most of the absorptivity, and a small direct chaotic set with a fractal boundary. The absorptivity of a whole run equals the probability that the first passage ends in excursion or in chaos (correlation of 0.98 in 147 situations), which reduces the problem to a single encounter; in it, the choice between flyby and exchange depends almost only on the masses, but there is not yet a universal law.

Zenodo (CERN European Organization for Nuclear Research)
Openalex Percentile: Top 16%
Quantum chaos and dynamical systems
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